Problem: What is the largest $4$ digit integer congruent to $15 \pmod{22}?$
Answer: An integer that is congruent to $15 \pmod{22}$ is of the form $22n+15$.

Therefore, we make the equation $22n+15<10000$, and find the largest possible $n$. \begin{align*}
22n+15&<10000 \\
22n&<9985 \\
n&<\frac{9985}{22} \approx 453.85
\end{align*}The largest possible integer $n$ is $453$. We plug it in for $n$ to get $22 \cdot 453 +15 =\boxed{9981}$.